Week 2 & 3: Computing truth values

Truth table

A B ---------------- 1. T T 2. T F 3. F T 4. F F

Modal logic

A kind of logic that deals with concepts such as possibility, necessity, belief, and doubt

Truth-functional proposition/sentence

A proposition is truth-functional when its truth value depends upon the truth values of its component parts.

Sub-formula

A sub-formula is a meaningful formula that occurs as part of another formula Ex: (~E v H ) ≡ (~G . F) sub-formulas are: E, H, G, F, ~E, ~G, (~E v H ), (~G . F)

Bivalence

Every proposition is either true or false

Truth table short cuts

If one disjunct is true, the entire disjunction is true. If one conjunct is false, the entire conjunction is false. If the antecedent is false, the conditional is true. If the consequent s true, the conditional is true.

Truth table for conditional

It is a little unintuitive. Any conditional with a false antecedent is always true. When the consequent is true, the conditional is also always true. The conditional is only false when the consequent is false, and the antecedent is true. p q | (p ⊃ q) ----------------- T T | T T F | F F T | T F F | T

Negative adjectives vs sentences

Negative adjectives such as "un-" work the same way as not and make a sentence compound. Ex: John is ungrateful = John is not grateful Unhelpful = not helpful

main operator

The main operator is the operator that determines the overall structure of a WFF (well-formed formula). The main operator cannot be the negation operator.

How do we use variables like p and q?

To indicate the conjuncts may be representing any formulas (simple of complex) Ex: (((A v B) ⊃ C). ~(D v E)) p q | (p . q) ----------------- T T | T T F | F F T | F F F | F

Importance of paratheses

We can use parentheses to eliminate ambiguities.

How to compute truth tables

Write the truth value of each simple sentence below its corresponding letter, and then proceed by computing the values of each operator, starting from these inside the parentheses till you have reached the main one.

Well-formed formula (WFF)

a grammatically correct symbolic expression.

antecedent

a thing or event that existed before or logically precedes another

Inclusive disjunction

allows both disjuncts to be true.

Non-truth-functional operator

an operator is non-truth-functional when we can come up with pairs of component sentences with identical truth value and show that by compounding them, they turn out to have a different truth value output.

exclusive disjunction

declares that only one or the other, but not both, of the disjuncts is true.

epistemic logic

examines concepts such as knowing, believing, thinking, and related operators

consequent

happening as a result of something

Truth table for inclusive "or"

p q | (p v q) ----------------- T T | T T F | T F T | T F F | F Always true unless both p and q are false.

Truth table for biconditional

p q | (p ≡ q) ----------------- T T | T T F | F F T | F F F | T Both p and q must be both true or both false to get a true value